Short of repealing the Second Amendment, the Federal gov't basically has no way of making Florida accept California's CCW laws.
On the other hand, if the feds passed a set of minimum concealed carry standards, whatever they may be, then every state would be required to accept the CCW license of any other state that meets those requirements.
This is how both driver's licenses and marriage licenses work right now. This is the reason that Texas does not have to recognize Massachusetts same-sex marriages---because marriages so licensed do not meet the Federal standards laid out in the Defense of Marriage Act.
If the Full Faith and Credit clause is properly interpreted, a basic set of federally legislated carry standards could only increase mandatory reciprocity, not decrease it, nor affect any individual state's own licensing procedures.